Sang-il Oum엄상일

Abstract

We show that the associahedron $\mathcal{A}_n$ has chromatic number $O(\log\log n)$, improving on the previously best known upper bound of $O(\log n)$.

Abstract

Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)-1}$ for $t\ge4$. We lower the base of the exponential by one: for every $t\ge5$, every $P_t$-free graph $G$ satisfies \[ \chi(G)\le 3\,(t-3)^{\omega(G)+4}. \] The proof combines two refinements of the Gyárfás path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.

Abstract

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.